Consider a planet $P$ of radius $r$ larger than Earth, whose mass is $r^3$. By the square–cube law, this planet will have the same density as the Earth... assuming that the gravitational compression on it is the same as on our planet, but this is wrong, because the planet will have a greater gravity, therefore, the gravitational compression in it will be more intense, making it smaller and denser than expected (is this reasoning correct?). But what matters to me is not only the planet's density, but also that of its core, which would also increase.

The Earth's core has a density of 11,000 kg/m³, greater than would be expected if it were calculated considering it only as a mixture of two substances (Fe and Ni, to simplify), without taking into account the gravitational compression. How can I calculate the density of the core of $P$ (and of itself) taking into account the gravitational compression? Is it possible to do it without much trouble?

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    $\begingroup$ There's no handy equation for the relationship between these things simply because real materials will not have constant density or temperature as radius (in the planet) changes. Here's a link to a page with links to various planet models which will give you some idea of the complexity. There's a (not easy to understand) application than can be used via Wolfram Player if you want to try it. I suspect this material will be too complex for you, but have a look. $\endgroup$ – StephenG Aug 23 '19 at 9:39
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    $\begingroup$ A tricky question because of phase changes: apparently under high pressure, iron at least can go from liquid back to crystalline solid. phys.org/news/2017-02-theory-earth-core-solid-extreme.html . Phase can definitely affect density. I propose this ? would be good for the planetary science stack since it is definitely hard science, but I don't know how to make it migrate. $\endgroup$ – Willk Aug 23 '19 at 13:03
  • $\begingroup$ Maybe the way Earth's density was mapped out? That included a lot of things like tracking earthquake waves, measuring exact values of gravity, etc. Knowing the value at the core is a messy thing without measurements. $\endgroup$ – puppetsock Aug 23 '19 at 13:42

The technique to do this is similar to that used in constructing stellar models. You know some of the properties of your object - in this case, it look like the mass and radius. You want to figure out the internal structure of the planet, including the central density and pressure, as well as the density and pressure profiles as functions of radius. The best way to do this is as follows:

  1. Determine a relationship between density and pressure - what we call an equation of state. We write this as $\rho = f(P)$.
  2. Make a guess as to the central pressure - and therefore the central density, using the above relation.
  3. Numerically integrate the differential equations determining the structure of the planet. Start in the core, and go outwards.
  4. Once your equations indicate that you've reached the surface of the planet, see what the predicted mass and radius are.
  5. Given the result of step 4, adjust your guesses for the central pressure and density, and repeat steps 3 and 4.

The two equations you need, besides the equation of state, are the equation of hydrostatic equilibrium $(1)$ and the equation of mass continuity $(2)$: $$\frac{dP}{dr} = -\frac{GM_r\rho}{r^2}\tag{1}$$ $$\frac{dM}{dr} = 4\pi r^2\rho\tag{2}$$ Here, $P$ is pressure, $\rho$ is density, $r$ is radius, $G$ is the gravitational constant, and $M_r$ is the mass contained within a radius $r$: $$M_r = \int_0^r 4\pi r'^2\rho(r')dr'$$ Let's say that at a given radius $r_i$, you know the pressure, density, and mass encapsulated within that radius - $P_i$, $\rho_i$, and $M_{r,i}$. Assume that your integration uses some step size $\Delta r$. Then the values at the radius $r_{i+1}$ are $$r_{i+1} = r_i + \Delta r$$ $$P_{i+1} = P_i + \Delta P = p_i + \frac{dP}{dr}\Delta r = P_i - \frac{GM_{r,i}\rho_i}{r_i^2}\Delta r$$ $$\rho_{i+1} = f(P_{i+1})$$ $$M_{r,i+1} = M_i + \Delta M = M_i + \frac{dM}{dr}\Delta r = M_i + 4\pi r_i^2\rho_i\Delta r$$ Continue doing this until you reach the surface of the planet, i.e. at $P = 0$, then repeat steps 3 to 4 with a new guess until you're satisfied with the model's predicted mass and radius.

I implemented this procedure in my answer to another question, although I only had to do steps 3 to 4 once - I had a reasonable central pressure all figured out. I used an equation of state from Seager et al. 2008, designed for rocky planets.

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    $\begingroup$ You need information that you don't have. You write $\rho(r)$ but it's a function of composition as well. You don't know that. $\endgroup$ – puppetsock Aug 23 '19 at 16:31
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    $\begingroup$ @puppetsock The OP has presumably chosen a composition (e.g. iron, perovskite, etc.). The linked paper gives sample parameters for various compositions (see Table 3), and it's quite simple to just pick the right ones depending on the planet's composition (or perhaps choose intermediate values if the planet has a more heterogeneous composition). It's not hard to find a proper $\rho(P)$ composition-dependent equation of state. $\endgroup$ – HDE 226868 Aug 23 '19 at 16:34
  • $\begingroup$ It's not hard to find a proper $\rho(P)$ composition-dependent equation of state It may not be hard for you but my impression is that this is very hard for the OP. $\endgroup$ – StephenG Aug 23 '19 at 20:54
  • $\begingroup$ @StephenG I linked to a paper with several different EOSs that cover most rocky planets. . . I'm not sure what the remaining objections are. If the OP has a specific composition, I can of course look for an appropriate one if it's not covered in the paper, but right now, I'm not sure if the current options are inadequate. :-/ $\endgroup$ – HDE 226868 Aug 24 '19 at 2:14
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    $\begingroup$ I'll update the answer with a more detailed section on EOSs within a couple of days. I think future readers might find that useful, beyond the OP, so I'll go into more detail when possible. $\endgroup$ – HDE 226868 Aug 24 '19 at 2:24

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